BHS 307 Statistics for the Behavioral Sciences

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BHS 307 Statistics for the Behavioral Sciences

• Chapter 6 Correlation

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Does Aleks Quiz 1 Predict Midterm Scores?

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Adding a Prediction (Regression) Line Provides
More Information
r .56
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Does Time Spent on Aleks Predict Quiz Grades?
r .16
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Sometimes the Relationship is Not Linear
r .16 r .47 (quadratic)
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Describing Relationships

• Positive relationship high values are paired
  with high values, low with low.
• Negative relationship high values are paired
  with low values, low with high.
• No relationship no regularity appears between
  pairs of scores in two distributions.

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Relationship Does Not Imply Causality

• A relationship can exist without being a CAUSAL
  relationship.
• Correlation does not imply causation.
• Third variable problem -- a third variable is
  causing both of the variables you are measuring
  to change.
• The direction of causality cannot be determined
  from the r statistic.

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Scatterplots

• One variable is measured on the x-axis, the
  other on the y-axis.
• Positive relationship a cluster of dots sloping
  upward from the lower left to the upper right.
• Negative relationship a cluster of dots sloping
  down from upper left to lower right.
• No relationship no apparent slope.

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Strength of Relationship

• The more closely the dots approximate a straight
  line, the stronger the relationship.
• A perfect relationship forms a straight line.
• Dots forming a line reflect a linear
  relationship.
• Dots forming a curved or bent line reflect a
  curvilinear relationship.

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Examples

• http//www.stat.uiuc.edu/courses/stat100/java/GCAp
  plet/GCAppletFrame.html

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Correlation Coefficient

• Pearsons r a measure of how well a straight
  line describes the cluster of dots in a plot.
• Ranges from -1 to 1.
• The sign indicates a positive or negative
  relationship.
• The value of r indicates strength of
  relationship.
• Pearsons r is independent of units of measure.

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Interpreting Pearsons r

• The value of r needed to assert a strong
  relationship depends on
• The size of n
• What is being measured.
• Pearsons r is NOT the percent or proportion of a
  perfect relationship.
• Correlation is not causation.
• Experimentation is used to confirm a suspected
  causal relationship.

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Calculating Pearsons r

• S zxzy r _______ n 1
• Computation formulas
• Different textbooks use different formulas.
• Aleks gives you a choice.
• Use whichever one is most convenient given the
  information in the problem.

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Dealing with Outliers

• Outliers can dramatically change the value of the
  r correlation coefficient.
• Always produce a scatterplot and inspect for
  outliers before calculating r.
• Sometimes outliers can be omitted.
• Sometimes r cannot be used.
• http//www.stat.sc.edu/west/javahtml/Regression.h
  tml

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Other Correlation Coefficients

• Spearmans rho (r) based on ranks rather than
  values.
• Used with ordinal data (qualitative data that can
  be ordered least to most).
• Point biserial correlation -- correlations
  between quantitative data and two coded
  categories.
• Cramers phi correlation between two ordered
  qualitative categories.